**5. Physical Explanation**

In this study, the dimensionless parameters that were appearing in the momentum and the energy equations and the value of these parameters were taken to be fixed for the computational purpose are given as λ = −3.5, *n* = 0.5, *We* = 0.5, γ1 = γ2 = 0.5, ε1 = 0.5, *B* = 0.1. The graphical features of the embedded flow of fluids were captured in Figures 2–21 on the velocity, temperature profiles, the skin friction, and the local Nusselt number against the enormous distinct parameters. The numerical results with accessible conclusions are referenced in Table 1, which shows the authenticity of our problem by comparing the results with the available results in the literature. Additionally, the green lines throughout the study demonstrate the first solution, which is also called the upper branch solution while the red lines exhibit the second solution called the lower branch in all the invoked figures.


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**Table 1.** Comparison of the values of *f*(0) for distinct values of ε1 and λ when *K* = 1 and rest of variables are absent.

### **6. Deviation of the Skin Coe**ffi**cient and the Local Nusselt Number**

The graphical behavior of our solutions for the skin friction coefficient 0.5(Re*x*) 2 (Pr) 2 *Cf* and the local Nusselt number *Nux*(*Pex*) −1 2 by exercising the different parameters against the mixed convection parameter λ are shown in all invoked Figures 2–7. The existing of dual solutions is marked in all the aforementioned figures in the case of mixed convection opposing flow (λ < 0) while the outcome is unique for the phenomenon of mixed convection assisting flow (λ > <sup>0</sup>). The influence of the modified porosity parameter ε1 on the skin friction and the local Nusselt number versus λ is depicted in Figures 2 and 3, respectively. Figure 2 shows that the values of the skin friction decelerate in the first solution with enhancing ε1 in the range of (−<sup>4</sup> ≤ <sup>λ</sup>), while the reverse trend is seen in the range of (λ < <sup>−</sup><sup>4</sup>). Figure 3 scrutinizes that the values of the Nusselt number accelerate due to ε1. It is also observed from these sketches that the physical realizable solution is represented by the green solid lines and the decline of the unstable solution is displayed by the red dotted lines. The critical values |λ| enhance as ε1 augments, suggesting that the modified porosity parameter delays the boundary-layer separation. In addition, it can be clearly observed from these figures that the skin friction as well as the Nusselt number augments as λ increases in the assisting flow, while the contrary behavior is observed in the opposing flow. Physically, in the assisting flow case, the favorable pressure gradient produces which augments the motion of liquid, which consequently raises the shear stress and heat transfer rate. In contrast, opposing flow guides to an adverse pressure gradient that delays the motion of liquid. The impacts of the Weissenberg number *We* and the inertia parameter *B* against λ on 0.5(Re*x*) 12 (Pr) −12 *Cf*

and *Nux*(*Pex*) −12 are depicted in Figures 4–7. For the upper branch solution, both the momentum boundary layer and the thermal boundary layer become lower by changing the value of *We*, while the opposite behavior is marked for the lower branch solution as shown in Figures 4 and 5. Figures 6 and 7 sugges<sup>t</sup> that the fall trend with augmenting *B* in the lower branch solution, while the upper branch solution is enhanced for the similar choice of *B*.

**Figure 3.** Influence of ε1 on *NuxPe*−1/2 *x* .

**Figure 5.** Influence of *We* on *NuxPe*−1/2 *x*.

**Figure 7.** Influence of *B* on *NuxPe*−1/2 *x*.

### **7. Deviation of the Velocity and Temperature Fields**

The analyses and the behavior are captured in Figures 8 and 9, respectively, showing the tasters of the *f*(η) and <sup>θ</sup>(η) profiles for the distinct values of the slip parameter γ1 for both branches of the solutions, while the effects of the thermal slip parameter γ2 on the velocity and temperature for various selected values are portrayed in Figures 10 and 11, respectively. Physically, when augmenting the values of γ1, the wall shear stress insignificant and as a result, the momentum boundary layer (Figure 8) becomes larger and larger for both the upper branch and lower branch solutions, while the reverse trend is scrutinized for the temperature profile (Figure 9) due to escalating the γ1. Figure 10 shows that the velocity of fluid rises with γ2 in the first solution and declines in the second solution, while the opposite behavior is observed in the sketch of temperature, as shown in Figure 11. This is due to fact that the extra flow penetrates through the thermal boundary layer which consequently transmitted the additional heat and this guides in the decline of temperature distribution. Thus, for the authenticity of our solutions, it is clearly visible from behavior of momentum and temperature profiles in Figures 8–11 that these solutions satisfied the boundary conditions asymptotically. As shown in Figures 12–15, the behavior of the fluid flow is explored by exercising the dimensionless parameters *We* and *n* on *f* (η) and <sup>θ</sup>(η), respectively. The increment in the local Weissenberg number *We* and power-law index *n*, both the green solid lines, as well as the red dotted lines (first and second solution) are rising in Figures 12 and 13, while the contrary flow of fluid motion is noticeable corresponding to these parameters in Figures 13 and 15, respectively. From the physical view, the additional relaxation time is needed when the values of *We* increases and as a result, the velocity boundary layer and the fluid temperature was shrunk and declined in Figures 12 and 13, respectively. Figure 14 exhibits that the velocity profile increases due to the augmenting values of *n* in case of shear thinning and vice versa for the temperature profile which is invoked in Figure 15. Figure 16 shows the behavior of the permeability parameter *K* on *f* (η) as we enhance the parameter *K*, the upper solution is decelerated while the lower solution shows increasing behavior, whereas for the same parameter, the reverse behavior is noted in the temperature profile, as presented in Figure 17. Figure 18 illustrates that the liquid velocity enhances in both upper and lower solutions by changing the values of the modified porosity parameter ε1, while the temperature profile behavior is shown in Figure 19, which decelerates in both branches of solutions as we boost up the value of ε1. In Figure 20, we plotted the velocity profile for various values of inertia coe fficient *B*, which shows that the first solution is enhanced and the second solution is declined. The temperature profile declines in the upper branch solution and rises in the second branch solution as the value of *B* augments and this behavior is captured in Figure 21. The cause for this trend is that the inertia of the porous medium o ffers an extra confrontation to the mechanism of the liquid flow, which grounds the liquid to progress at a retarded rate with reduced temperature.

**Figure 8.** Influence of γ1 on *f* (η).

**Figure 10.** Influence of γ2 on *f*(η).

**Figure 12.** Influence of *We* on *f*(η).

**Figure 14.** Influence of *n* on *f*(η).

**Figure 16.** Influence of *K* on *f*(η).

**Figure 18.** Influence of ε1 on *f*(η).

**Figure 20.** Influence of *B* on *f*(η).

**Figure 21.** Influence of *B* on <sup>θ</sup>(η).
